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In the context of regression-type statistical models, the inclusion of some ordered factors among the explanatory variables requires the conversion of qualitative levels to numeric components of the linear predictor. The present note represent a follow-up of a methodology proposed by Azzalini (2023} for constructing numeric scores assigned to the factors levels. The aim of the present supplement it to allow additional flexibility of the mapping from ordered levels and numeric scores.

Stress and material deformation field predictions are among the most important tasks in computational mechanics. These predictions are typically made by solving the governing equations of continuum mechanics using finite element analysis, which can become computationally prohibitive considering complex microstructures and material behaviors. Machine learning (ML) methods offer potentially cost effective surrogates for these applications. However, existing ML surrogates are either limited to low-dimensional problems and/or do not provide uncertainty estimates in the predictions. This work proposes an ML surrogate framework for stress field prediction and uncertainty quantification for diverse materials microstructures. A modified Bayesian U-net architecture is employed to provide a data-driven image-to-image mapping from initial microstructure to stress field with prediction (epistemic) uncertainty estimates. The Bayesian posterior distributions for the U-net parameters are estimated using three state-of-the-art inference algorithms: the posterior sampling-based Hamiltonian Monte Carlo method and two variational approaches, the Monte-Carlo Dropout method and the Bayes by Backprop algorithm. A systematic comparison of the predictive accuracy and uncertainty estimates for these methods is performed for a fiber reinforced composite material and polycrystalline microstructure application. It is shown that the proposed methods yield predictions of high accuracy compared to the FEA solution, while uncertainty estimates depend on the inference approach. Generally, the Hamiltonian Monte Carlo and Bayes by Backprop methods provide consistent uncertainty estimates. Uncertainty estimates from Monte Carlo Dropout, on the other hand, are more difficult to interpret and depend strongly on the method's design.

Estimating parameters of a diffusion process given continuous-time observations of the process via maximum likelihood approaches or, online, via stochastic gradient descent or Kalman filter formulations constitutes a well-established research area. It has also been established previously that these techniques are, in general, not robust to perturbations in the data in the form of temporal correlations. While the subject is relatively well understood and appropriate modifications have been suggested in the context of multi-scale diffusion processes and their reduced model equations, we consider here an alternative setting where a second-order diffusion process in positions and velocities is only observed via its positions. In this note, we propose a simple modification to standard stochastic gradient descent and Kalman filter formulations, which eliminates the arising systematic estimation biases. The modification can be extended to standard maximum likelihood approaches and avoids computation of previously proposed correction terms.

The ability to manipulate logical-mathematical symbols (LMS), encompassing tasks such as calculation, reasoning, and programming, is a cognitive skill arguably unique to humans. Considering the relatively recent emergence of this ability in human evolutionary history, it has been suggested that LMS processing may build upon more fundamental cognitive systems, possibly through neuronal recycling. Previous studies have pinpointed two primary candidates, natural language processing and spatial cognition. Existing comparisons between these domains largely relied on task-level comparison, which may be confounded by task idiosyncrasy. The present study instead compared the neural correlates at the domain level with both automated meta-analysis and synthesized maps based on three representative LMS tasks, reasoning, calculation, and mental programming. Our results revealed a more substantial cortical overlap between LMS processing and spatial cognition, in contrast to language processing. Furthermore, in regions activated by both spatial and language processing, the multivariate activation pattern for LMS processing exhibited greater multivariate similarity to spatial cognition than to language processing. A hierarchical clustering analysis further indicated that typical LMS tasks were indistinguishable from spatial cognition tasks at the neural level, suggesting an inherent connection between these two cognitive processes. Taken together, our findings support the hypothesis that spatial cognition is likely the basis of LMS processing, which may shed light on the limitations of large language models in logical reasoning, particularly those trained exclusively on textual data without explicit emphasis on spatial content.

Precision matrices are crucial in many fields such as social networks, neuroscience, and economics, representing the edge structure of Gaussian graphical models (GGMs), where a zero in an off-diagonal position of the precision matrix indicates conditional independence between nodes. In high-dimensional settings where the dimension of the precision matrix $p$ exceeds the sample size $n$ and the matrix is sparse, methods like graphical Lasso, graphical SCAD, and CLIME are popular for estimating GGMs. While frequentist methods are well-studied, Bayesian approaches for (unstructured) sparse precision matrices are less explored. The graphical horseshoe estimate by \citet{li2019graphical}, applying the global-local horseshoe prior, shows superior empirical performance, but theoretical work for sparse precision matrix estimations using shrinkage priors is limited. This paper addresses these gaps by providing concentration results for the tempered posterior with the fully specified horseshoe prior in high-dimensional settings. Moreover, we also provide novel theoretical results for model misspecification, offering a general oracle inequality for the posterior.

Noninformative priors constructed for estimation purposes are usually not appropriate for model selection and testing. The methodology of integral priors was developed to get prior distributions for Bayesian model selection when comparing two models, modifying initial improper reference priors. We propose a generalization of this methodology to more than two models. Our approach adds an artificial copy of each model under comparison by compactifying the parametric space and creating an ergodic Markov chain across all models that returns the integral priors as marginals of the stationary distribution. Besides the garantee of their existance and the lack of paradoxes attached to estimation reference priors, an additional advantage of this methodology is that the simulation of this Markov chain is straightforward as it only requires simulations of imaginary training samples for all models and from the corresponding posterior distributions. This renders its implementation automatic and generic, both in the nested case and in the nonnested case.

Due to their flexibility and theoretical tractability Gaussian process (GP) regression models have become a central topic in modern statistics and machine learning. While the true posterior in these models is given explicitly, numerical evaluations depend on the inversion of the augmented kernel matrix $ K + \sigma^2 I $, which requires up to $ O(n^3) $ operations. For large sample sizes n, which are typically given in modern applications, this is computationally infeasible and necessitates the use of an approximate version of the posterior. Although such methods are widely used in practice, they typically have very limtied theoretical underpinning. In this context, we analyze a class of recently proposed approximation algorithms from the field of Probabilistic numerics. They can be interpreted in terms of Lanczos approximate eigenvectors of the kernel matrix or a conjugate gradient approximation of the posterior mean, which are particularly advantageous in truly large scale applications, as they are fundamentally only based on matrix vector multiplications amenable to the GPU acceleration of modern software frameworks. We combine result from the numerical analysis literature with state of the art concentration results for spectra of kernel matrices to obtain minimax contraction rates. Our theoretical findings are illustrated by numerical experiments.

Mechanical issues of noncircular and asymmetrical tunnelling can be estimated using complex variable method with suitable conformal mapping. Exsiting solution schemes of conformal mapping for noncircular tunnel generally need iteration or optimization strategy, and are thereby mathematically complicated. This paper proposes a new bidirectional conformal mapping for deep and shallow tunnels of noncircular and asymmetrical shapes by incorporating Charge Simulation Method. The solution scheme of this new bidirectional conformal mapping only involves a pair of linear systems, and is therefore logically straight-forward, computationally efficient, and practically easy in coding. New numerical strategies are developed to deal with possible sharp corners of cavity by small arc simulation and densified collocation points. Several numerical examples are presented to illustrate the geometrical usage of the new bidirectional conformal mapping. Furthermore, the new bidirectional conformal mapping is embedded into two complex variable solutions of noncircular and asymmetrical shallow tunnelling in gravitational geomaterial with reasonable far-field displacement. The respective result comparisons with finite element solution and exsiting analytical solution show good agreements, indicating the feasible mechanical usage of the new bidirectional conformal mapping.

Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion or diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary Partial Differential Equations containing such operators and integrated in time with exponential integrators, it is then of paramount importance to efficiently approximate the actions of $\varphi$-functions of the arising matrices. In this work, we show how to produce directional split approximations of third order with respect to the time step size. They conveniently employ tensor-matrix products (the so-called $\mu$-mode product and related Tucker operator, realized in practice with high performance level 3 BLAS), and allow for the effective usage of exponential Runge--Kutta integrators up to order three. The technique can also be efficiently implemented on modern computer hardware such as Graphic Processing Units. The approach has been successfully tested against state-of-the-art techniques on two well-known physical models that lead to Turing patterns, namely the 2D Schnakenberg and the 3D FitzHugh--Nagumo systems, on different hardware and software architectures.

For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for the multi-dimensional variable-order fractional Laplacian defined by a hypersingular integral. We prove that the scheme is of second-order convergence and apply the developed finite difference scheme to solve various equations with the variable-order fractional Laplacian. We present a fast solver with quasi-linear complexity of the scheme for computing variable-order fractional Laplacian and corresponding PDEs. Several numerical examples demonstrate the accuracy and efficiency of our algorithm and verify our theory.